Question

Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. then $P^2{R}^3:S^3$ is equal to

• A. 1 : 1
• B. $($Common ratio${)}^n:1$
• C. $($ First term ${)}^2$ $($ Common ratio ${)}^2$
• D. None of these

Answer 1

The correct option is A

1 : 1

(a) 1 : 1

Let us take a G.P. with three terms $\frac{a}{r},a,ar.$ Then,

$S=\frac{a}{r}+a+ar=\frac{a(r^2+r+1)}{r}$

$P=a^3,R=\frac{r}{a}+\frac{1}{a}+\frac{1}{ar}=\frac{1}{a}\left(\frac{r^2+r+1}{r}\right)$

$\frac{P^2{R}^3}{S^3}=\frac{a^6.\frac{1}{a^3}{\left(\frac{r^2+r+1}{r}\right)}^3}{a^3{\left(\frac{r^2+r+1}{r}\right)}^3}=1$

Therefore, the ratio is 1 : 1